Critical and ramification points of the modular parametrization of an elliptic curve

Critical and ramification points of the modular

parametrization ^of

^an

elliptic

^curve

par CHRISTOPHE DELAUNAY

RÉSUMÉ. Soit E une courbe

elliptique

^définie sur Q ^{de conduc-}

teur N et soit ~ ^son revêtement modulaire :

~ ^:

Xo (N)

^~

E (C) .

Dans cet article, ^{nous nous} intéressons aux points critiques ^et

aux points de ramification de _~. En

particulier,

^nous

expliquons

comment donner une étude

plus

^{ou moins}

expérimentale

^{de ces} points.

ABSTRACT. Let E be an

elliptic

^{curve defined}

over Q

^{with con-}

ductor N and denote

by

~ the modular parametrization:

~ :

Xo (N)

^~

E(C) .

In this _paper, we are concerned with the critical and ramification

points ^of ~. In

particular,

^we

explain

^{how we can obtain a more}

or less

experimental study

^{of these} points.

1. Introduction and motivation

1.1. The modular

parametrization.

^{Let E be an}

elliptic

^{curve defined}

over Q

^with conductor N. It is known from the work of

Wiles, Taylor

^and

Breuil, Conrad, Diamond, Taylor ([16], [14], [3])

^{that E is modular}

and, hence,

that there exists a map cp, called the modular

parametrization:

where

Xo(N)

^{is the}

quotient

^{of the extended} upper half

plane

^{H = H U}

by

^the congruences

subgroup Fo (N) (which

^we endow with its natural structure of

compact

^Riemann

surface).

^The

analytic

group

isomorphism

from

(C/A

^to

E((C)

^is

given by

^the

Weierstrass p

^{function and its derivative}

and we will often

implicitly identify

^these two spaces. We now

briefly

describe the _{map cp.}

The

pull-back by

cp of the

holomorphic

differential dz is

given by:

where

f (z)

^{is a}

(normalized)

newform of

weight

^{2 on}

ro(N)

^{and c E Z is}

the Manin’s constant. In this _paper, we assume that we have c ⁼

1;

ⁱⁿ

fact all

examples

^of

elliptic

^{curves we have studied}

satisfy

^this

assumption (we

^{use the first}

elliptic

^{curves from the} very

large

table constructed

by

Cremona

[7]).

Since the differential

2i-xf (z)dz

^is

holomorphic,

^the

integral:

does not

depend

^{on the}

path

^{and defines a}

rnap §5 :

^C.

Furthermore, if l’ E ro(N),

^{we set:}

the _{map w} does not

depend

^{on T and is called a}

period

^of

f (see [6]

to

compute

^it

efficiently).

^The

image

^{of cv :}

C,

^{which is a} group

homomorphism,

^{is a lattice A c C and}

then,

^we

get

^a map:

The curve E is in fact chosen

(en

^even

constructed)

^{such that}

E(C)

and _cp is the modular

parametrization.

^This

description

^{of the modular}

parametrization

^{is very}

explicit

^{and allows us to evaluate} cp at ^a

point

T E

Xo(N),

^indeed:

o

First,

suppose that ⁷ is not a cusp, then we have :

where the

(integers) a(n)

^are the coefficients of the Fourier

expansion

^of

f

^{at the} cusp ^oo. Note that

they

^are

easily computable

^since

L( f, s) =

¿n a(n)n-S

⁼

L(E, s)

^is the L-function of E. The series

(1.1)

^{is a}

rapidly converging

^{series and}

gives

^{an efficient} way to calculate _cp.

~

Suppose

^{now that 7}

represents

^a cusp in

Xo(N). Then,

^{the series}

(1.1)

does not converge any ^more and we have to use a different method. Let be the "modular

symbol"

^{as described in}

[6] (it represents

^a

path linking

oo to T on which we will

integrate f ).

^{The Hecke}

operator Tp,

^{where p is a}

prime number,

^{acts on modular}

symbols

^{and we have:}

mod N then the _{cusps p7} and

(T + j ~ ~p (for j

⁼

0, ... ,

p -

1)

^are

equivalent

^{to T mod}

To(N),

^{so there exist}

Mo, Ml,

^...,

Mp

^E

To(N)

^such

that:

Writing ~(T ~- j)/p} _ {(T + j)/p~ -

⁺

{T~

^and

using

^{the fact that}

f

^is

a Hecke

eigenform

^with

eigenvalue a(p)

^for

Tp,

^{we obtain:}

And we deduce from it the value of the _{map cp at} the _cusp T

(of

course, this is a torsion

point

^{and with the method}

above,

^{we can}

compute

^it

exactly

and not

only numerically).

1.2. Critical and ramification

points.

^The

topological degree of cp

^can

be

efficiently computed by

several methods

([6], [15], [17], [8]),

^{and we}

denote

by

^{d =}

deg(cp)

^this

integer.

^{There are}

finitely

many

points z

^E

E(C)

^{for which}

~~cp-1(~z})~

^d

(we

^{will call z a} ramification

point).

These

points

^are

image

^{of critical}

points (and

^{so critical}

points

^{are finite}

in

number).

^{We are} interested in

localizing

^and

studying

^the ramification and the critical

points.

^{From the Hurwitz} formula there are

exactly 2g - 2

critical

points (counted

^with

multiplicity) where g

is the genus of

Xo(N)

and is

given by:

...

In this

formula,

p ⁼

~SL2(7G) :

^{voo is} the number of _cusps and _v2

(resp. v3)

^{is the number of}

elliptic points

^{of order 2}

(resp.

^order

3)

^of

Xo (N) .

^Critical

points

^{are the zeros of the} differential

dcp

⁼

2i7rf (z)dz

^and

then we are led to find all the zeros of

f . Nevertheless,

^{a zero of}

f

may ^not be a critical

point

^{because dz has}

only

^a

polar part ([12]):

where Pj

^{are the cusps} and ej

(resp. e’)

^the

^elliptic

elements of order 2

(resp.

^order

3)

ⁱⁿ

Xo(N). Thus,

^{c E}

Xo(N)

^{is a critical}

point

^for cp if and

only

^{if c is a zero of}

f

^{not due to a}

pole

^{of dz. For}

example,

^a

cusp P

^is critical if and

only

^{if the order of}

vanishing

^of

f

^{at P is} > 2.

Note that the differential form

f (z)dz

^is

holomorphic

^{and so a}

pole

^{of dz is}

counterbalanced

by

^{a zero of}

f . Thus,

^the modular form

f

^{vanishes at all}

elliptic

^{elements of}

Xo(N). Furthermore,

^{with the usual} local coordinates for

Xo(N),

^the

multiplicity

^{of a zero of}

f

^{at an}

elliptic

element of order 2

(resp.

^order

3)

^{has to} be counted _up with the

weight 1/2 (resp. 1/3).

So,

^a

simple (resp. double)

^{zero of}

f ,

^{viewed as a function}

f : C,

at an

elliptic

^{element of order 2}

(resp.

^order

3) compensates exactly

^the

corresponding pole

^{of dz}

(viewed

ⁱⁿ

Xo(N)).

Let c E

Xo(N)

^{be a critical}

^point

of the modular

parametrization,

^so

cp(c)

^{is a} ramification

point

^{and is also an}

algebraic point

^on

E(Q)

^defined

over a certain number field K.

Then,

^the

point

^{is a rational}

point

^on

E(~).

^{A natural}

question

^{asked in}

[11]

^{is to} determine the sub- group

generated by

^{all such rational}

points. They

^denote

by

^this

subgroup.

^{A critical}

point

^{c E}

Xo(N)

^{is said to be a} fundamental critical

point

^{if c E}

zR,

^{we denote}

by

^the

subgroup

^of

E(Q) ^{generated by}

where c runs over all the fundamental critical

points.

^In

~11~,

the authors

proved

^the

following:

Theorem 1.1

(Mazur-Swinerton-Dyer).

^The

analytic

^rank

of

^{E is less}

than or

equal

^to the numbers

of fundamental

^critical

points of

^odd

order,

and these nurrcbers have the same

parity.

In this _paper, we are concerned with

studying

^these

particular points

of the modular

parametrization.

^More

precisely,

^{for a fixed}

elliptic

^curve

E,

^the

questions

^{and the}

problems

^{we are} interested in are to determine the critical

points,

^{to define a} number field K in which live ramification

points,

^{to look at} the behaviour of the

subgroups E(Q)"’

^and

E(Q)f°’"d,

etc..

Although

it is clear that these

questions

^{can be}

theoretically

^answered

with a finite number of

algebraic (but certainly

unfeasible in the

general case) calculations,

^the

part

^{3 and 4 of our work is more or less}

experimental

and some of the answers we

give

^{come from numerical}

computations

^{and are}

not

proved

^{in a}

rigorous

way.

Nevertheless,

^{there are some}

strong

^evidence

to be

enough

self confident in their

validity.

2. Factorization

through

Atkin-Lehner

operators

In order to find

(numerically)

all the critical

points

^of cp, ^we will

explain,

in the next

section,

^{how to solve the}

equation f (z)

⁼

^{0, z}

^E

Xo(N).

^In

fact,

the results we have obtained

by

^{our method}

suggest

^{that the zeros of}

f

^{are often}

quadratic points

ⁱⁿ

Xo(N).

Let

WQ (with Q N and gcd(Q, N~Q)

⁼

1)

^{be an} Atkin-Lehner

operator.

Since

f

^{is an}

eigenform

^for

WQ,

^{we have}

flwQ ^{= + f,}

^{we deduce that:}

Furthermore,

^{2P = 0 if the}

sign

^{is +1.}

Suppose that p

^o

WQ

^{= cp, then we}

can factor the modular

parametrization through

^the operator

W~ :

The _map 7rQ ^is the canonical

surjection,

^its

degree

^{is 2 and we have}

deg(cp)/2.

^{The fixed}

points

^{c of}

WQ

ⁱⁿ

Xo(N)

^{are critical}

points

^for 7rQ and

thus are critical

points

^for cp. The Hurwitz formula

gives:

where 9Q ^is the _genus of

Xo(N)/WQ.

^{We define a function}

Hn(A)

^{for n E N}

and A 0.

If n ⁼

1, Hl(0)

^{= is the} Hurwitz class number of A

(cf. [5]).

If n >

2,

^{we write =}

a2b

with b

squarefree,

^{and we let:}

For n E

N,

^{we denote}

by Q(n)

^the

^greatest integer

^{such that}

Q(n)2 ~ n, by Qo(n)

the number of

positive

^{divisors of n and}

by p(n)

the Moebius function.

Using

the results of

[13],

^{we have:}

Proposition

^{2.1. Let}

tr(WQ, ^S2(N))

^{be the trace}

^of

^the

^operator WQ

ⁱⁿ

the _space

S2(N) of

^the cusp

forms of weight

^{2 on}

ro(N),

^then:

where:

The numbers gQ and

tr(WQ, ^S2(N))

^{are related}

^by:

and so the number of fixed

points

^of

WQ

ⁱⁿ

Xo(N)

^is

given by

^{the formula:}

We are

looking

for all fixed

points

^of

WQ .

^{We let:}

and we search T E 1HI such that

WQT

^{= MT with M E}

^Changing

the

coefficients

y, z et w of

WQ,

^{we can assume that M =} I d and that

= T.

Then,

^{the matrix}

W~

^is

"elliptic"

^and:

First,

suppose that

Q fl 2,

^{3 then z = -w and:}

Furthermore

Q

⁼

det(WQ) = _Q2X2

⁺

^yNz

^{so we have}

(2Qx)2

⁼

-4Q

⁺

4Nyz

^{and the}

point:

is a fixed

point

^of

WQ. Conversely,

^{we can} check that _every fixed

point

^has

the form

(2.2).

^{In order to} find all the fixed

point:

. We search

/3 (mod 2N)

^{such that:}

~ For each divisor z of +

4Q)/4N,

^we

get

^{the fixed}

point:

For each new T, ^we check that it is not

equivalent

^mod

ro(N)

^{to a}

point already

found. We

try

^{a new}

divisor z, eventually

^we

change

the solution

f3

^{and we continue. We}

stop

^{when we have} obtained all the fixed

points (their

^{number is known}

by

^formula

(2.1)).

The critical

point

^{T in formula}

(2.4)

^has

precisely

^{the form of an}

Heegner point

of discriminant

C~’

^{for some} space

Xo(N’)

^with

^Q’

^{and N’ convenient}

(see ~4~, [9]

^{for more} details about

Heegner points).

^In

fact,

^{with the nota-}

tions above:

~ If 2 N and 2

y

^and

2 t z

^{then T is an}

Heegner point

^of discriminant

-Q

^{in the} space

Xo(Q~.

~ If 2 N and 2

y

^{and 2}

f z

^{then T is an}

Heegner point

of discriminant

-4Q

^{in the} space

Xo(Q).

~ Otherwise T is an

Heegner point

^of discriminant

-4G~ (-Q

^{if 2}

y

and 2 in the _space

Xo(N).

If T is a critical

point

^for cp and an

Heegner point

of discriminant

A,

^then

write A ⁼ where A’ is a fundamental discriminant and

f

^{is the}

conductor of the order 0 in the

quadratic

^{field K = In this} case, the ramification

point cp(T)

^{is defined over} the number field H ⁼

the

ring

^class field of conductor

f.

If

Q

^{= 2 or}

3,

^we

have I x + w~ I

^{= 0}

or I x

⁺

w[ I

^{= 1. The first case is}

discussed

above,

^{for the}

second,

^{we have to}

adapt

^the method with w ⁼ 1 - _~; this

changes nearly nothing.

^The

equation (2.3)

^is

replaced by:

And the fixed

points

^are:

where A ⁼ -4

(resp.

^{A =}

-3)

^whenever

Q

^{= 2}

(resp. Q

⁼

3).

Sometimes,

^the Atkin-Lehner

operators

^{does not suffice to factor} cp and

we can also have to use other

operators;

^those

coming

from the normalizer in

SL2(R) ([I]).

^In

^practice,

^the

computations

^are

analogous.

Example.

^{Let take E defined}

by:

Its conductor is N ⁼ 80 and the _genus of

Xo (80) is g

⁼

7,

furthermore

deg(p)

^{= 4 and we have to find 12 critical}

points

^for cp.

First of factors

through

^the Atkin-Lehner

operator W16

^{which has}

4 fixed

points

ⁱⁿ

Xo(N).

The method above allows us to determine them:

The operator:

belongs

^{to the} normalizer of

To (N)

^{and we have cp o W =} cp. The operator W is an involution of

Xo(N) (we

^{can not}

simplify

^{it because}

W2

^and

S2

^do

not

commute).

^{There are 4 fixed}

points

^{of W which are:}

In

fact,

cp is

completely

factorized:

The critical

points

^{of 7r2 are} ci, c2, c3 et C4- We have C5 ⁼

7r2(C5) _ 7r2(C7)

and _C6 ⁼

7r2(C6) == 7r2(CS),

^{so in}

Xo(80)/W2:

Thus, 1r2(C5)

^and

7r2(c6)

^{are critical}

points

^for

7r2.

^{Let denote}

by Pl, P2, P3 and P4

^the

following

^different cusps of

Xo(N):

- -- --

We have

7r2(Pl) == IP1, P2}

^and

7r2 (P3) = P3, P4 1.

^The

operator

^{W acts}

on these _cusps

by WPl

⁼

P2

^and

WP,3

⁼

P4.

We deduce that and

7r2 (P3)

^{are critical}

^points

^{for W.}

Finally,

^we have obtained the 12 critical

points

^for cp and the ramification

points

^are:

Definition 1. An

elliptic

^{curve E is}

involutory if

^{there exists}

operators Ul, U2, ..., Uk belonging

^{to the} normalisator

of Fo(N)

ⁱⁿ

SL2(R) through

which _cp can be

completely factorized :

and where

Xj

⁼

Xj-l/Uj

^and

Uj

^{is an} involution

of Xj-l.

The curve of the

example

^is

involutory

and the critical and ramification

points

^{can be}

completely

described. In the table

(1),

^we

give

^a list off all

involutory

^{curves E} with conductor N 100 such that E is not

isomorphic

to

Xo(N).

The column

(Uj)j gives,

^with

order,

involutions for a

complete

factorization of _cp.

The factorizations

through operators (completely

^or

partially) explain

most of the time

why

^{some critical}

points

^are

quadratic.

Nevertheless there

are cases for which critical

points

^are

quadratic

^{whereas no} factorization

seems to be

possible.

This is the _case, for

example,

^{for the curve E defined}

by:

-

TABLE 1.

Involutory

^{curves with conductor N}

100,

^and

such that

deg( cp) =1=

^{1. The} curves are

given

^{in the form}

y2 +

^{alxy + a3y}

^{= ~3}

^-f-

^{a2~2 -t~}

^{+ as.}

with conductor N ⁼

33,

the 4 critical

points

^{are the}

following Heegner points:

We have

deg(p)

^{= 3 and it is hard to}

explain

^this

by

^{a natural}

operator.

If one wants to prove that a

quadratic point

^{is a zero of}

f ,

one can use the

following

way.

Let _{~1,~2? ... ,} l’ J-L be ^{a set} of

representatives

^{for the}

right

^cosets

of Fo (N)

in

SL2 (Z) ,

^{then classical}

arguments

^{show that the} coefficients of:

belong

^to

Z[j] (at

^{least if N is}

squarefreee), where j

^is the modular invariant and A the usual _cusp form of

weight

^{12 on} the full modular _group. The

computation of P(X)

^is

possible (but

delicate and

long

^{since the} coefficients

explode

^with

N).

Consider the constant term of this

polynomial P(0) _ G(j)

^{where G is a}

polynomial

^with

integer

coefficients. We have:

Let _zo be a

quadratic point. Then, j(zo) belongs

^{to the}

ring

^of

integers

^{of a}

certain number field and so

belongs

^{to some discrete}

subgroup

^of

C,

^hence

it is _{easy to} determine if

G(j(zo))

^{= 0.}

Furthermore,

^the factorization of

G(x)

ⁱⁿ irreducible elements

gives

^{the order of}

vanishing at j(zo). Now, by

numerical

computations,

^{we can decide}

which qj

^{are such that}

^0,

the

others qj

^{lead to zero of}

^f (A

^{does not vanish in}

H).

Coming

^{back to the}

example

^{of the}

elliptic

^{curve E} with conductor N ⁼ 33. We let

f(T) = q + q2 - q3 _ q4 _ 2q5

^{+ ... be the} newform of

weight

²

on

ro(33)

associated to the

elliptic

^{curve E.}

Then,

^{in this} case, ^we have:

And the

points j(ci), ’’’ ~(04),

^where cl, - - - c4 ^are defined

above,

^{are the}

zeros of this

polynomial.

3. Localization of the zeros of

f

Whenever a zero of

f

^{is neither a}

quadratic point

^on

Xo(N)

^{nor a cusp}

then it is a transcendental number in H

(because j (z)

^{and z are both}

algebraic

^{if and}

only

^{if z is}

quadratic).

^{In this}

^section,

^we propose to

give

^a

method in order to calculate at least

numerically

^{the zeros of}

f

^{and hence}

the critical and ramification

points

of the modular

parametrization.

Let:

Then,

^we

complete

^{the set}

{ao, ... , by

matrices aN+ 1, ^{... ,} a. ⁱⁿ order to obtain a set of

representatives

^{for the}

right

^{cosets of}

To(N)

ⁱⁿ

SL2(Z) .

i.e.:

If N is

prime,

^{we need not}

complete

^{the first set}

since,

^{in this} case, ^we have We let:

be the classical fundamental domain for the full modular _group

SL2(Z).

Then,

^{we define:}

where

WN

⁼

0

^is the Fricke

involution,

^{is a} fundamental domain for

Xo(N).

^It may be not connected. We will search the zeros of

f

1T1

Proposition

3.l. There exists a

positive

real number B ⁼

Bf

^{0.27 such}

that:

.1 . I I -

, ^{1 1}

Proof.

^We

apply

^{Rouch6’s theorem to the function q H}

En

^D.

The

(very bad)

^{bound B 0.27 comes from 2n. D}

For

applications

^we

compute

^{a more accurate numerical value for}

B,

ⁱⁿ

fact B seems to decrease as N _grows. In

[11],

^{we say that a cusp}

^a/b

^is

a

unitary

cusp if

gcd(b, N/b)

^{= 1}

(in

^this

definition,

^we

implicitly

suppose that the

representative a/b

is such that

gcd(a, b)

^{= 1 and}

biN).

Corollary

^{3.1. Let}

a/b

^{be an}

unitary

cusp, there exists a

neighborhood

V C

Xo(N) of a/b

^{such that:}

Proof. If a/b

^is

unitary,

^{one can find an} Atkin-Lehner operator ^{W such that}

Wa/b =

oo, and we take V ⁼ >

B}).

^D

This also _proves that an

unitary

cusp is never a critical

point

^for cp; in

fact,

^{one can also} prove this

by determining

^{the Fourier}

expansion

^of

f

^at

the _cusp

a/b.

^If

a/b

^{is not}

unitary,

^then

a/b

may be a critical

point (see

the

example above).

^At

least,

^{we remark that:}

Thus,

^{we can}

always

^{choose z’ E such that and we}

can

forget

the half-disk

fz

^E

H , Izi

^{when we are}

looking

^for

critical

points.

All the considerations above allow us to restrict the domain of search for the zeros of

f (see figure 1).

FIGURE 1. Fundamental domain for

Xo(N) (and restrictions)

Then,

^{in the domain}

left,

^we localize the zeros of

f using Cauchy’s

^inte-

gral.

^More

precisely,

^we

partition

^{the domain in small}

parts

^{R and for each} R we

compute:

It is an

integer:

^{if it is zero then R does not contain} any ^zero of

f ,

^otherwise

R contains some zeros.

Then,

^{we divide R in small}

parts

^{and continue.}

When the localization of a zero is

enough precise,

^{we use} Newton method in order to obtain the desired _accuracy.

Once the zeros of

f obtained,

^{we check that}

they

^are

inequivalent (be-

cause the Newton ’s method _may

jump

^{a zero out} of the restricted

domain).

We eliminate those

coming

^{from the}

elliptic points.

^{There must have}

2g - 2

critical

points

^{left. The} method above is

quite

^{efficient in}

practice,

^even

in the case where zeros are

multiple.

^{Let denote}

by

ci, c2, ^{... ,} C29_2 the critical

points

^of p.

Then,

^we consider the ramification

points

associated:

zj ^{= =}

(xj, yj).

^{From the}

^algebraic properties

^{of the} zj ^the

polyno-

mials :

have rational coefficients. We can compute ^them

numerically

^{and we}

verify

that the results we obtain are closed to certain rational

polynomials;

^this

provides

^{a check of}

computations. Indeed,

^{in each case we have}

considered,

we

get

^a

polynomial

very close to a rational

polynomial

^{and we can then}

be

enough

self confident in the results.

Furthermore, P~

^and

Py

^{enable us}

to define a number field K where the

points zj

^{are defined.}

Example

^{Let consider the}

elliptic

^curve of conductor N ⁼ 46 defined

by

the

equation:

It is the first curve

(for

^the

conductor)

^for which the critical

points

^{are not}

Heegner points.

^The genus of

Xo(46) is g

^{= 5 so there are} 8 critical

points.

We determine them

by

^{the method}

explained

^above:

And

then,

^{we can}

compute 7~.

^and

Py (numerically):

There are 93

isogeny

^{classes of}

elliptic

^{curves with conductor N 100.}

Among them,

^{46 are}

involutory

^{and are listed in table}

(1) (more precisley

those which are not

isomorphic

^to

Xo(N)).

^{There are 30 non}

involutory

curves for which all the critical

points

^are

quadratic (so

^{due to some fac-}

torization and "accidental" reason as for the case N ⁼

33).

^Some cusps

are critical

points

^for 4 of the 93 _curves,

they

^{are all}

involutory (N

⁼

48,

N ⁼ 64 and the 2 curves with N ⁼

80).

^{The first case for which the}

group is not of finite index in

E(Q)

^{is obtained}

by

^{the curve}

y2 ~- y

⁼

^{~3 +~2 -}

7~+ 5 with conductor ^{N = 91. For} this curve

E( Q)crit

^is

of finite index. The

curve y2 ^{= 3 -}

^{x + 1 with conductor N = 92 is the first}

example

^{of curve with rank 1 for which}

E(Q)crit

^{is a torsion}

sub-group.

4. A rank 2 curve

In this

section,

^{we comment some of the results we obtained}

concerning

the first

elliptic

^{curve of rank 2 over}

Q.

^{This is the}

elliptic

^{curve with}

conductor N ⁼ 389 defined

by:

The group

E(Q)

^{is torsion free of rank 2}

generated by ^G1

⁼

(0, 0)

^and

G2

⁼

(1, 0).

^{The genus}

of Xo(389) ^{is g}

⁼

^32,

^{there are 2 cusps}

(389

^is

prime)

and no

elliptic

^element.

Furthermore, deg(p)

^{= 40. We}

computed

^{the 62}

critical

points

^of cp. As

predicted by

^theorem

1.1,

^{there are 2} fundamental critical

points:

More

astonishing,

^{2 critical}

points

^are

Heegner points

of discriminants -19:

Two critical

points belong

^{to the line}

Re(s)

⁼

1/2:

The last remark comes from a

generalization

of theorem 1.1:

Theorem 4.1. Let r be the

analytic

^rank

of E

^{and m be the number}

of

critical

points of

^{odd order}

lying

^{on the line}

1/2

⁺

~

If

²

t

^{N or 4 ~}

N,

^{then r m and these numbers have the same}

parity. If a(2)

^{= 2 we have the}

stronger inequality

^{r + 2 m.}

~ then r m and these numbers have the same

parity if

^and

only if a(2) =

^-1.

Proof. First,

^{the end}

points

^{of the line I =}

1/2

^{+ zR are not critical}

^point

since

f

^has

only

^a

single

^{zero at the} cusp

1/2 (indeed,

^{if 4}

1

^{N then}

f(T

⁺

1/2) _ - f (T),

^if

21 IN

^then

1/2

^{is a}

^unitary

^{cusp and so a}

single

zero of

f

^{and if 2}

~’

^{N the} cusp

1/2

^{and oo are}

equivalent).

^A direct calcula- tion also shows that there are not any

elliptic point

^{on I}

(because

N >

11),

thus all zeros of

f

^{on I n IHI are critical}

points

^for cp.

If 4

1

^{N then}

f (T

⁺

~) ^{_ - f (T)}

^{and the}

proposition

^{comes from theo-}

rem 1.1.

Otherwise,

^{let we have:}

where

L(E, s)

^{is the} L-function of E

(or f )

^and

L2(E, X)

^{is its} Euler factor at p ⁼ 2. One can see that the order of

vanishing v

^of

2L2(E, 2-s) -

^{1 at}

s =0 is:

~ v ⁼ 2 if

a(2)

^{= 2}

(in

^this case, ^we have 2

{ N).

. v ⁼ 0 otherwise.

Then the order of

vanishing

^of

(the analytic

continuation

of ) L*(E, s)

^at

s ⁼ 1 is r’ ⁼ r + v.

Furthermore,

and we can follow the

proof

^{of Mazur and}

Swinnerton-Dyer.

Since the order of

vanishing

^of

L* (E, s)

^{at s = 1 is r’ then:}

Let

1 /2 ~- itl,

^{... , the zeros of}

f

^{of odd order}

lying

^{on 7DBL}

Then,

is of constant

sign,

^{and the}

integral:

J ^‘

is not zero.

But,

^this

integral

^{is a linear} combination of

Jo,

^{... ,}

Jr,.t

^{and so} r’. This _proves the

inequalities

^of the theorem.

If N is odd then the Fricke involution:

/ - , - -,. ,

maps the line

Re(s)

⁼

1/2

^{to itself. Since}

f(z)

^{= 0 O}

f(WNz)

^{= 0 with}

the same order of

vanishing,

^the

parity

^of the number of zero of odd order of

f lying

^{on I n H is}

given by

^the

parity

^{of the order of}

vanishing

^of

f

^at

the fixed

point

^of

WN (i.e.

^{at T =}

1/2~). ^But, flwN

⁼

^-Ef,

^{where E is}

the

sign

^{of the} functional

equation

^of

L(E, s),

^{so we see} that this order has the same

parity

^{than the}

analytic

^{rank of E.}

If

211N

^{then the} Atkin-Lehner involution:

I --,- - I - I- 1

maps the line I to itself. The same

argument

^as before allows us to con-

clude except ^{that in this case we have}

flwN/2 == ^{a(2)e f}

^which

^explains

^the

eventual difference between the

parities.

^D

For our curve E with conductor N ⁼

389,

^{we have}

cp(c3)

⁼

cp(c4)

^{= 0 so}

we have determined the

polynomial ~*(X -~(cp(c)))

^{where the}

product

^runs

through

^{the critical}

points ^except

^{c3 and c4 and where}

denotes the x-coordinate of P. The numerical

computations

^{of lead}

to a

polynomial

^{which is closed to a rational}

polynomial, giving

^{a check of}

computations.

^This

polynomial

^{is the} square of an irreducible

polynomial over Q

^of

degree

^{30 with} denominator:

It also _appears that the

sub-group

^{is a} torsion group.

Acknowledgement.

^{Parts of this work has been written}

during

^{a Post-}

doctoral

position

^{at the}

Ecole Polytechnique

F6d6rale de Lausanne. The author is _very

pleased

^{to thank the}

prof.

^Eva

Bayer

^{and the members of}

the "Chaire des structures

alg6briques

^et

g6om6triques"

^{for their}

support

and their kindness.

References

[1] ^A.O.L. ATKIN, ^J. LEHNER, ^Hecke operators ^on 03930(N). ^{Math. Ann. 185} (1970), ^134-160.

[2] ^C. BATUT, ^K. BELABAS, ^D. BERNARDI, ^H. COHEN, ^M. OLIVIER, pari-gp, ^{available at}

http://www.math.u-psud.fr/~belabas/pari/

[3] ^C. BREUIL, ^B. CONRAD, ^F. DIAMOND, ^R. TAYLOR, On the modularity of elliptic curves over

Q: ^{wild 3-adic} exercises. J. Amer. Math. Soc. 14 (2001), ^{no. 4, 843-939} (electronic).

[4] ^B. BIRCH, Heegner points of elliptic ^curves. Symp. Math. Inst. Alta. Math. 15 (1975),

441-445.

[5] ^H. COHEN, ^{A course in} computational algebraic ^number theory. Graduate Texts in Math. 138, Springer-Verlag, New-York, 4-th corrected printing (2000).

[6] ^J. CREMONA, Algorithms for ^modular elliptic ^curves. Cambridge University Press, (1997)

second edition.

[7] ^J. CREMONA, Elliptic ^{curve data} for conductors up to ^{25000. Available at}

http://www.maths.nott.ac.uk/personal/jec/ftp/data/INDEX.html

[8] ^C. DELAUNAY, Computing ^modular degrees using L-functions. Journ. theo. nomb. Bord. 15

(3) (2003), ^673-682.

[9] ^B. GROSS, Heegner points ^on X0(N). ^Modular Forms, ed. R. A. Ramkin, (1984), ^87-105.

[10] ^B. GROSS, ^D. ZAGIER, Heegner points ^and derivatives of L-series. Invent. Math. 84 (1986),

225-320.

[11] ^B. MAZUR, ^P. SWINNERTON-DYER, Arithmetic of ^{Weil curves.} Invent. Math. 25 (1974),

1-61.

[12] ^G. SHIMURA, Introduction to the arithmetic theory of automorphic functions. ^{Math. Soc of} Japan 11, ^Princeton university ^Press (1971).

[13] ^N. SKORUPPA, ^D. ZAGIER, ^Jacobi forms ^{and a} certain space of ^modular forms. ^{Inv. Math.}

98 (1988), 113-146.

[14] ^R. TAYLOR, ^A. WILES, Ring-theoretic properties of certain Hecke algebras. ^{Ann. of Math.}

(2) ¹⁴¹ (1995), ^no. 3, ^553-572.

[15] ^M. WATKINS, Computing the modular degree. Exp. ^{Math. 11} (4) (2002), ^487-502.

[16] ^A. WILES, ^Modular elliptic ^{curves and} Fermat’s last theorem. Ann. of Math. (2) ¹⁴¹ (1995),

no.3, 443-551.

[17] ^D. ZAGIER, ^Modular parametrizations of elliptic ^{curves. Canad. Math. Bull. 28} (3) (1985),

372-384.

Christophe ^DELAUNAY Institut Camille Jordan Bâtiment Braconnier

Universite Claude Bernard Lyon ¹ 43, ^avenue du 11 novembre 1918 69622 Villeurbanne cedex, ^France

E-mail : delaunay@igd.univ-lyon1.fr